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Combinations vs. Permutations - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

So this is a problem where we are looking at both combinations and permutations, so there is these two issues that seems to have trouble with distinguishing between or else we're going to throw them together to look at them side by side.

So what we have is a school government is comprised of a president, a vice president and a secretary and then you have 4 senators and we have 20 students who are interested in government, how many different governing bodies are possible?

So basically what we're saying is there are 20 students it doesn't matter who takes what position, they all want to be part of it somehow and we're trying to figure out how many different combinations we can have. So the first thing we want to do is to choose our president, vice-president and secretary and so we have 20 students and we need to choose these 3 important roles.

The first thing we have to figure out is if it is a combination or a permutation, does order matter? If our president and vice-president switch, do we have a different governing body? The answer is yes, so we have order matters which tells us we are dealing with a permutation.

So the first thing we want to do is we take 20 and we permute 3, so that's going to take care of our president, vice-president and secretary, order matters so we need to permute this. So we're now left with 17 students we took care of 3 of them and now we have 17 left. So we're going to need to multiply by 17, to do something and then we are doing 4, we now need to figure out if it's this doing here is a permutation or a combination.

For our senators, it doesn't always say they have any specific role any different than another, so if we interchange one senator from the other, we're still going to end up with the same senate body so in this case order doesn't matter which tells us we are dealing with a combination. So what we have left is 20 permute 3 is first going to give us our 3 sort of officials and then 17 choose 4 is going to give us that pool of 4 people who are interchangeable.

So a combination of permutation with combinations hopefully you can see the difference with the permutation our order matters with choosing our order doesn't.

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